DOCUMENT RESUME

ED 055 913 SE 012 625

TITLE Tenth Year Mathematics.INSTITUTION New York State Education Dept., Llbany. Bureau of

Secondary Curriculum Development.PUB DATE 71NOTE 32p.; Reprint from the syllabus Mathematics

10-11-12

EDRS PRICEDESCRIPTORS

MF-$0.65 RC-$3.29Algebra; Arithmetic; *Curriculum Guides; *Geometry;*Grade 10; Instruction; *Mathematics; TeachingProcedures

ABSTRACTThe booklet presents the minimum material for which

students are responsible on the Tenth Year Regents examination of thestate of New York. The syllabus is an attempt to integrate planegeometry with arithmeticl algebra and numerical trigonometry broughtabout by: (1) greater use of fractions and percents in mensurationproblems; (2) use of approximate number; (3) use of algebraicsymbolism and proof; (4) use of algebraic equations in the solutionof geometric problems; and (5) an introduction to coordinategeometry. The scope of content is: transition from informal to formalgeometry, formal geometry (triangles, inequality, parallelism andperpendicularity, angle sum, locus, circles, angle measurement,similarity, areas, regular polygons and measurement of circles)constructions, formulas, arithmetic, algebra, trigonometry, andcoordinate geometry. Suggestions for teaching, for time schedule andfor sequencing are included. (JG)

U S 'EFARTMENT OF HEALTHEOUDATION 8. WELFAREOFFICE OF EDUCATIQN

THIS DOCUMENT HAS BEEN REPRO-

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IONE. STATED DO NOT NECESSARILY

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CA ;ION POSITION OR pOLICY

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TNE WHVERSITY:OFTHE STATE OFTNEW_YORK/THE

STATE EDUCATION DEPARTMENT

BUREAU'OiSECONDARY-CURRICULUMDEVELOPMENT/ALDANY,

NEW YORK 1222ii,

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TENTH YEAR MATHEMATICS

LU

Reprint from the syllabus, Ma 2

1971

The University of the State of New YorkThe State Education DepartmentBureau of Secondary Curriculum DevelopmentAlbany, New York 12224

_

THE UNIVERSI1Y OF THE STATE OF NEW YORK

Regents of the University (with years when terms expire)

1984 Joseph W. McGovern, A.B., LL B. L.H.D., LL.D., D.C.T..Chancellor

1985 Everett J. Penny, B.C.S., D.C.S.,Vice Chancellor

1978 Alexander J. Allan, Jr., LL.D., Litt.D.

New York

White Plains

TroyA1973 Charles W. Millard, Jr., A.B., LL.D., L.H.D. - --- Buffalo

1972 Carl H. Pforzheimer, Jr., A.B., M.B.A., D.C.S., H.H.D. Purchase1975 Edward M. M. Warburg, B.S., L.H.D. New York1977 Joseph T. King, LL.B. Queens1974 Joseph C. Indelicate, M.D. Brooklyn1976 Mrs. Helen B. Power, A.B., Litt.D., L.H.D., Rochester1979 Francis W. McGinley, B.S., LL.B., LL.D. Glens Falls1980 Maz.c. J. Rubin, LL.B., L.H.D.

X986 Kenneth B. Clark, A.B., M.S., Ph.D., Litt.D.

1982 Stephen K. Bailey, A.B., B.A. M.A., Ph.D.,1983 Harold E. Newcomb, B.A.

1981 Theodore M. Black, A.B.

President of the University and Conwissioner of Educationwald B. Nyqul

uty CaGorden M M a hExecutive De

oina

issiener of Educatien

ssionerfor Elementary; Sec _d

Associate issioPhilip B. Lan- orth

and Cant nuin

New York

Hastings onHudson

Syracuse

Owego

Sands Point

Educat' n

atructional Services

Assistant Commission r for 1 structiGnal Services (General Education)Bernar.

-Director, -DiviSion of chool:Su.ervis n -eorge Mu

Chief Bureau of 'Seconda Cur iculu- Davie ep-enGer on E. Van Ho

Di eater Division of General EducationT . T Grena

Chief, Bureau of Mathematics EducationFrank orno

FOREWORD

This publication is a reprint of the Tenth Year Matiematiossection of the syllabus, Mathematics 10-1112, An Integrated Sequencefor the Senior High Sehooi Grades. It presents thr minimum materialfor which students are responsible on the Tenth Year Regents exam-ination.

A revision of the tenth year course is underway to accompanythe separate publications Ninth Year MathaMatics-Course 1, Algebra(1965), EleVenth Year Mathematics (1968), and Ex.pei,imentai 22thYear Mathematics (1060).

Additional suggestions for teaching varius topics may befound in the Mathematics Handbook, A Handbook of Resource Materitc. Accompany tha Course of Study in Mathematics 10 (2962).

Copies of the above publications may be ordered by the principalfrom the Publications Distribution Unit, Sta Education Departmen,Albany, New York 12224.

OEF 1ITION

While no immediate changes in the scope, fermatsor coverage of theTenth Year Mathematics examinations are contemplated, certain changes ofnotation and language will be made beginning with the June 1969 examinationand continuing until further notice. This does not involve a change in thepresent syllabus which will be followed in regard to topics tested until aformal revision is prepared and distributed.

The following notation and language will be used:

Line Representation(1)

AB means line segment AB

AB means the length of line segment AB

AB means the line determined by A and B

ABC means the line segment AC with point B between A and C

(2) Measure

m L ABC means the measure of k ABC in degrees

(3)

m AB means the measure of arc AB in degrees

Congruence-

The symbol will be used to designate line segments of equal lengths,angles of equal measure, circles of equal radii, ores of -equal measureon circles of equal radii, as well as congruent polygons.

The symbol as appliedto such Figures means they are identical.

The language and notation_of the Ninth Year Mathematics _yllabus may,-course, also be used where appropriate,

The afereMentioned changes:are intended to make the questions on the'eXamination more nearly precise and to Make the language,of the'exaMinationconsistent with modern usage:employed in many schools.'. Students will be expectec,to recognize this symbolism hut need not follow it in their answers.

It is emphasized that this is a change of language and n_ ation and is n tto be construed as a change in the basic syllabus for the course. Thesechanges in language and notation are being made with the advice and concur-rence of the committee of teachers composing the Tenth Year Mathematics ReganExaminations

TENTH YEAR MATHEMATICSFor many years courses in plane geometry have consisted largely

of a body of sorucwhat unrelated geometric theorems, the performingof simple geometric constructions, the solUtion of original exercises,an introduction to the study of locus and a brief reviev., of thenumerical trigonometry of the ninth grade. Little or no attention hasbeen given to the use or to the extension of the basic principles ofarithmetic and algebra. As a result, many of these concepts and skillsare lost during the tenth year from lack of use. One purpose of thepresent syllabus is to remedy this situation by attempting to integrateplane geometry with arithmetic, algebra and numerical trigonometryin so far as such integration is possible and desirabie. Some of theways in which this may be brought about are:

Greater use of common and decimal fractions and per cents inmensuration problems

An introduction to the meaning and use of approximate numberIncreased emphasis on numerical trigonometryThe use of algebraic symbolism and algebraic proof wherever this

is desirableThe use of algebraic equations in the solution of geometrIc problemsAn introduction to coordinate geometry

Another major change in this syllabus is the reduction of the num-ber of required geometric theorems- concentration on a few groupsof closely related theorems may be expected to result in a clearerunderstanding of the nature of proof and the meaning of sequentialthinking. Furthermore, under former syllabuses there has alwaysbeen the tendency to memorize theorems and to drill on certain typesof original exercises. It is hoped that reducing the number oftheorems required for examination purposes will discourage this.

While it is generally agreed that formai logic should not be a partof the tenth grade work, it is possible to develop some of the simpleideas related to this subject. Accordingly, the syllabus stressescertain points which may contribute to the pupil's ability to thinkcritically in nonmathematical as well as mathematical situations-Topics which can be related to affairs of everyday living are the sig-nificance of definition, the necessity for certain assumptions in any

12 THE UNIVERSITY Or THE STATE OF NEW YORK

argument, the nature of indirect proof, the recognition of converse andinverse theorems and the meaning of circular reasoning.

Perhaps the most conspicuous feature of the new syllabus is theaddition of a unit of coordinate geometry. In addition to the intrinsicvalue of the unit as a part of the mathematical program, it also makespossible an ideal integration of algebra and geometry and lends varietyand interest to the work of the tenth grade. This work is a continua-tion of that done in the ninth grade and provides preparation for workof greater difficulty and of greater value in the eleventh grade.

A suggested time schedule and teaching sequence accompani s thissyllabus, but this does not imply that either must necessarily befollowed. These are merely suggestions which may be of value tothose who have had little or no experience in teaching the course.

Scope of ContentI The transition from info mal to farma1 geometry

A Definition and use of basic terms and concepts (1)B The use of axioms and postulates (2)C Fundamental theorems (3)D Congruence (4)

Fundamental c nstructions (5)

II ormal geometryBasic proposstions

In view of the large number of propositions that maybe taught profitably, it can not. be expected that pupilsreproduce the proofs of all. It is suggested that from thefollowing list those numbered,1,-2, 3, 10; 11, 12, 13,,14, 30,31, 35, 36, 43, 44, 50, 54, 59, 66 be accepted without proof.

. -The other propositions given in This section should be_ proved in,class, but only those marked with an asterisk maybe called for on the exa.'mination.

Triang esTwo triangles are _,congruent sides and the

included,-angle of one are equal to the correspondingparts of the other.

thesis to Sugestion3 fo 'reaching on

MATHEMATICS, GRfiDES 10, 1 1, 12 13

2 Two triangles are congruent if two angles and theincluded side of one are equal to the correspondingparts of the other.

3 Two triangles are congruent if the three sides of oneare equal to the three sides of the other.

4 When two straight lines intersect, the vertical anglesare equal.

*5 If two sides of a triangle are equal, the angles oppositethese sides are equal.

*6 If two angles of a triangle are equal, the sides oppositethese angles are equal.

*7 Two right triangles are congruent if the hypotenuse anda leg of one are equal to the corresponding parts ofthe other_

Inequalitywo sides of a triangle are unequal, the angles oppositethese sides are unequal and the greater angle liesopposite the greater side.

9 If two angles of a triangle are unequal, the sIdesopposite these angles are unequal and the greater sidelies opposite the greater angl

10 An exterior angle of a triangle is greater than eithernonadjacent interior angle.

Parallelism -nd PerpendicularityThe perpendicular is the shortest line that can be

drawn from a given point to a-given line.12 If two lines are parallel to the same line, they are p ra lel

to each other.13 When two lines are cut by a transversa

alternate-interior (or- correspondingequal the two lines are parallel."hen two para:ilel lines ar cut by a transversal, thealternate-interior - (or corresponding) angles areequal.

15 The opposite sides of a parallelogram are equal.16 The diagonals of a parallelogram_ bisect each othe17 If the opposite sides -of a quadrilateral are equal,_ _

figure is a parallelogram.

nd a pair ofangles are

A.8 If two sides -of a quadrilateral arc equal and paral el.the figure is a parallelogram.

14 THE UNIVERSITY OF" THE STATE OF NEW YOZ-7;

19 If the diagonals of a quadrilateral bisect each other, thefigure is a parallelogram.

20 If three or more parallel lines cut off equal segmentson one transversal, they cut off equal segments onany transversal.

21 The line segment that joins the midpoints of two sidesof a triangle is parallel to the third side and equal toone-half the third side.

Angle SumThe sum of the angles of a triangle is equal to a straight

angle.The exterior ansie of a triangle is equal to the suns of

the two nonadjacent interior angles.24 The sum of the interior angles of a polygon of n sid s is

equal to (ss 2) straight angles.25 The sum of the exterior angles of a. polygon formed by

extending eaCh of its sides in succession is equal tostraight angles.

26 The locus of points equidistantis the perpendicular bisectorjoining the two points.

The locus of points within anfrom the sides of the angle

--Circles:

angle_ - --

The perpendicular bisectors of the sides of a trianglemeet- in a point which is equidistant from the three

_ _v-ertices.

he bisectors of the angles_ of a triangle_ meet in a pointwhich is equidistant from_the sides of the triangle.

two giv n pointsthe line segment

angleis the

and equidistantbisector of that

30 If in the same or equal circles tchords are equal.

31 If in the same -or equal circlesarcs are equal their

chords are= equal,_- their_ arcs are equal.

A diameter perpendicular to a chord of a. circle bisectsthe chord and its arcs.

If in the same, or in equal circles_two chords are equal.they are equidistant from the center.

MATHE ATICS, GRADES 10, 11, 12 15

34 If in the same or equal circles two chords are equidis-tant from the center, the chords are equal.

35 A line perpendicular to a radius at its outer extremityis tangent to the circle.

36 A tangent to a cire.e is perpendicular to the radiusdrawn to the point of contact.

37 Tangents drawn to a circle from an external point areequal.

38 Two parallel lines intercept equal arcs on a circle.

Angle Measurement*39 An angle inscribed in a circle is measured by one-half

its intercepted arc.40 An angle formed by a tangent and a chord drawn from

the point of contact is measured by one-half theintercepted arc.

*41 An angle formed by two chords inte secting inside thecircle is measured by one-half the sum of the inter-cepted arcs.

*42 An angle formed by two secants, a tangent and a secantor two tangents is measured by one-half the differ-ence of the intercepted arcs.

43 A line parallel to one side of a triangle and intersectingthe other two sides divides those sides proportionally.

If a line divides two sides of a triangle proportionally, itis parallel to the third side.

*45 If the three angles of one triangle are equal to the threeangles of another triangle, the triangles are similar.

46 If an angle of one triangle is equal to an angle of another'jangle and the sides including these angles are in

proportion the triangles are similar.*47 If in a right triangle the altitude is Arawn upon the

hypotenuse,The two triangles thus formed are

given triangle and similar to eachEach leg of the given triangle is the mean propor-

tional between the hypotenuse and the projectionthat leg on the hypotenuse.

15 THE UNIVERSITY OF THE STATE OF NEW YORK

Areas

48 If in a right triangle the altitude is drawn upon thehypotenuse, the altitude is the mean proportionalbetween the segments of the hypotenuse.

*49 The square of the hypotenuse of a right triangle is equato the sum of the squares of the legs.

50 If the square of one side of a triangle is equal to thesum of the squares of the other two sides, the triangleis a right triangle.

If two chords intersect within a circle, the product ofthe segments of one is equal to the product of thesegments of the other.

52 If from a point outside a circle a tangent and a secantare drawn the tangent is the mean proportionalbetween the secant and its external segment.

53 The perimeters of two similar polygons have the sameratio as any two corresponding sides.

54 If two polygons are similar, they can be divided into thesame number of triangles similar each to each andsimilarly placed.

*55 The area of a parallelogram is equal to the product oone side and the altitude drawn to that side.

*56 The area of a triangle is equal to one-half the productof a side and the altitude drawn to that side.

*57 The area of a trapezoid is equal to one-half the productof the altitude and the sum of the bases.

58 The areas of two similar triangles are to each otheras the squares of any two corresponding sides.

59 The areas of two similar polygons are to each other asthe squares of any two corresponding sides.

Regular Polygons and the Measurement of the Circle60 A circle can be circumscribed about, or inscribed in,

any regular polygon.Regular polygons of the same number of sides are

similar.The area of a regular polygon is equal to one-half the

product of its perimeter, and its apothem.The circumferences of two circles are to each other as

their radii.

61

*62

MATHEMATICS, GRADES 10, 11, 12 17

64 The ratio of the circumference of any circle to itsdiameter is constant. The constant is denoted byand hence c = 2.2Tr.

65 The area of a circle is equal to one-half the product 43its circumference and its radius. Hence A =

66 The areas of two circles are to each other as the squaresof their radii.

B Fundamental constructions1 To bisect a line segment2 To bisect an angle3 To construct a line perpendicular to a given line through

a given point on or outside the line4 To construct a line parallel to a given line through a

given point5 To divide a line into any number of equal parts6 To construct a line tangent to a given circle thro gh

a given point on or outside the circle7 To locate the center of a given circle8 To construct a circle inscribed in or circumscribed about

a given triangle9 To construct a triangle similar to a given triangle on a

given line segment as base10 To inscribe an equilateral triangl , a square and a regu-

lar hexagon in a given circle

C FormulasIn addition to the relatio ships derived from listed theo-

rems, such as, 24, 40, 48, 51, 53 and 58, certain mensurationmulas are to be taught intensively. Among these are theowing

(8)

Lines1 Right triangle L C = rt2 Equilateral triangle

Square ..4 Circ

18 THE UN1VERSITY OF THE STATE OF NEW YORK

Areas1 Rectangle K bh2 Square .. K =3 Parallelogram .. . K = bh4 Triangle K = bh5 Rhombus K = dd'6 Equilateral triangle 2

4

7 Trapezoid . K = b b'

8 Regular polygon K = aft9 Circle K -2T-r2

10 Sector of- a circle.. K 3602

D Proofs of original exerciseE Locus and construction

(9)(10)

II- ArithmeticA Use of integers, common and decimal fractions, and per

cents in mensuration problems (11)B Introduction to the meaning and use of approximate nutn-

bers (12)

IV AlgebraA The use of s gned numbers and the fundamen al process

of algebra as they occur in the applications of geometryEvaluation, transformation, and interpretation of formulas

relating to geometric figuresReview of radicals to cover the techniques needed for geo-

metric applications at this levelD Square root. Ability to use a table of square roots and an

understanding of the square root process.Ratio and proportion. Review and extension to cover all

geometric applications at this level.F Solution of equations. Review and extension to include the

solution of the compkte quadratic by factoring. (13)The Use of algebraic symbolism in geometric proof (14)Algebraic proof (15)

7-f="1-VVZ.,f-L;71:-Lt--1

MATHEMATICS, GRAES 10, 11, 12 19

V TrigonometryA The meaning of the sine, cosine.ane. tangent ratios resulting

from the study of similar trianglesB The use of a four-place table of sines, cosines and tangents.

Interpolation not required.C Solution of the right triangleD Problems in indirect measurement involving the use of ii

than one right triangleE Problems involving the regular polyeonF Development and use of the formulas K = ab sm C and

K ab sin C for the area of the parallelogram andtriangle respectively (optional ) (16)

VI Xntroduction to coordinate georne r (17)A Meaning and use of the terms : coordinate sy tem, axes of

reference, origin, abscissa, ordinateB Coordinates of points used in connection with problet.

involving loci and area (18)C Midpoint of a line segment (19)D Distance between two points (20)E Formula for the slope of a straight line (21)F Equations of straight lines (optional) (22)

Suggestions for Teaching1 Many of the basic terms and concepts of geometry have been

considered informally in grades 7-9 and certain fundamental rela-tionships have been derived informally. The pupil should now bemade to realize that such conclusions were reached largely throughmeasurement -and observation and hence are lacking in generality.Moreover, there is no evidence of relationship among theorernsreached by the experimental method alone. At this pol at there isneed for consideration of the nature of deductive proof illustrating theway in which some propositions follow,inevitably_from others. Thispermits the organization of many seemingly unrelated ideas into alogical organization called a postulational system-

_

20 THE UNIVERSITY OF THE STATE OF NEW YORK

In making the transition from informal geometry to formal geome-try, the pupil should he made to realize the importance both ofunderstanding clearly the basic concepts of geometry and of explain-ing these concepts by means of verbal statements. Eater these defini-tions play an important part in demonstrative geometry. In gen-eral, it is good practice to iritroduce only those terms and conceptsthat can be used immediately. At this stage it might be well to con-fine the discussion to those that are directly associated with points,lines, angles, polygons and circles. This work should include simplegeometric exercises of an experimental nature involving importantrelationships which are to be demonstrated later.

It is particularly important that the pupil appreciate that themaking of a definition consists of two parts, first, the placing of theconcept in a larger class of concepts (previously defined) and second,the assigning to this concept those particular characteristics whichdistinguish it from all other members of that class. Such practiceeasily brings about a realization of the importance of sequence in defi-nition and naturally leads to an understanding of the significance ofpropositional sequence. This is one of the main objectives of tenthgrade mathematics. Likewise, the need for undefined terms shouldbe made clear and this, in turn, associated with the necessity ofassumptions (axioms and postulates) in formal geometry. Atten-tion also should be given to the meaning of redundancy in definitionand to the fact that while redundancy is not always an undesirablefeature of a definition, it is better, generally speaking, to include in adefinition only the descriptive characteristics which are essential. Thisidea will be clarified later when the student is able to appreciate thedifference between a definition and a proposition, for example, thedefinition of a parallelogram as contrasted with the statements ofpropositions 17, 18, 19. Important, too, is the matter of the reversi-bility of a definition. This point, of course, can not be made clearuntil the pupil learns how to use definitions in formal proofs. Thereversibility of a definition, however, will be considerably clarifiedwhen the meaning of a converse proposition is understood.

he following exercises illustrate types I that may be usedmaking the transition from informal to formal geometry.

s -an aid

he inclu ion of certain illustrative- exercises throughout this syllabus mustnot be construed to mean that other exercises of greater difficulty or of differentkind may not appear on the final examination.

MAThEMATICS, GRADES 10, 11, 12 21

a Algebraic(1) The sum of the comple ent and the supplement of a

certain angle is 160'. Find the number of degreesin the angle.

(2) In the accompanying figure, BE is perpendicularto BD, LABE = x ± 2°, LEBD = 2xZDBC v 9°. Is ABC a straiht line ?

ricaw a triangle having three unequal sides. Using the

protractor measure the angles of the triangle. Whatconclusion concerning the sides and the opposite anglesseems warranted ?

Draw an isosceles triangle having an acute angle for thevertex angle. Measure the other two angles. Repeatthis with an obtuse angle and also with a right angle.What conclusion seems to be warranted concerningthe other two angles in each triangle ?

e eral(1) Define each of the following terms and arrange them in

proper sequential order : Perpendicular lines, rightangle, angle, adjacent angles.

(2) In the case of each of the following explain why the-

statement does not constitute a good definition :(a) A triangle is a polygon having three sides and three

angles.(b) A straight angle is an angle whose sides lie in the

same straight line.) A diagonal of a quadrilate al divides the quadri-

lateral into two triangles.

22 THE UNIVERSITY OF THE STATE OF NEW YORK

2 In the transition from informal to formal geometry the workshould be done slowly and carefully. Pupils should be made to appre-ciate the need for basic assumptions in any kind of argument, whethermathematical or nonmathematical. Care should be taken to selectonly those axioms and postulates of which immediate use can bemade. It is good practice to introduce the work by showing theapplications of the axioms of equality to the solution of simple equa-tions. Sufficient pains should be taken to insure that studentsunderstand the use of axioms in geometric settings. For example :If L a -= L b and L c = L d, what angle in the figure representsZ c? Lb Zd? Why is a c equal to Lb L d?

This work must include a sufficient number of simple geo_ etricproofs based on the use of definitions, axioms and postulates. Thework should begin with proofs having only one step, followed bythose having two, and then three or more steps.

3 After the nature of the basic assumptions is thoroughly under-stood and the student is able to prove these simple geometric exer-cises, cPrtain preliminary theorems may well be considered. Thesetheorems may be prescribed by the textbook or by the teacher. Theproofs of some of them may be assumed if it seems wise to do so ;others perhaps, should be demonstrated as class exercises. Forexample:

a Complementsare equal.

o adjacent angles have their exterior sides in the same -raight line the angles are supplementary.

straight lines intersect, the vertical angles are equal.

b I

upplements he sari o _qual augI

In any case these preliminary theorems, together with the defini-tions, axioms and postulates considered thus far, should be set apartas constituting a list of accepted authori -es which the student may

MATHEMATIcS, GRADES 10, 11, 12 23

use in proving original exercises. The distinction between what thestudent may use and may not use as reasons in proving originalsalways causes difficulty and, therefore, should be settled at this earlystage (see note 9, page 27). This work, too, should be followed byappropriate exercises and due attention should be given to the mannerin which proofs should be presented both orally and in written form.

4 The three fundamental laws of congruence (see propositions 1-3)may be developed informally as class exercises. This part of thework should include a number of simple geometric exercises in whichthe pupil learns how to prove that triangles are congruent and thatline segments and angles are equal. For example

a In isosceles triangle ABC, points R and S are the mid-points of legs AB and AC respectively. Line segmentsBS and CR are drawn. Prove triangle ABS congruentto triangle ACR.

b Triangles ACB and ADB have the same base AB and areon the same side of AB. Side AD intersects side BC at O.If AC = BD and BC = AD, prove that L CAD = L DBCand that CO = DO.

5 Emphasis here should be placed on the proofs of geometric con-structions (see note 7, page 26). Since the proof of proposition 5is usually based on the existence of the bisector of an angle, it maybe desirable to include proposition 5 at this point.

For the reason that the construction of geometric figures exemplifiesso well the meaning of the terms determined, underdetermined, andoverdetermined, it is recommended that a discussion of these ideas betaken up at this time. For example, in constructing a line througha given point perpendicular to a given line, the pupil frequently locatestwo additional points thus ON erdetermining the required line. Thiswork should be continued throughout other units as the necessarymaterials become available.

6 A proper study of geometry centers around certain broad topicssuch as congruence, construction, inequality, parallelism, angle sums,locus, angle measurement, similarity, measurement of plane figuresetc. It is fitting that the syllabus stress certain propositions fromthese fields and also their logical interrelationships within each field.The concept of the nature of proof as well as the logical structureexhibited in a chain of propositions are essential objectives in theteaching of formal geometry.

There are many ways in which the concept of sequential th nkingmay be taught. A recognition of the definitions and assumptions on

THE UNIVERSITY OF THE STATE OF NEW YORK

which the sequence is based, the logical order of the theorems whichmake up the seAquence, the possible interchange of certain theorems ina sequence without destroying the validity of the reasoning are someof the points to be stressed in presenting this feature of geometu.

In order that these objectives may be more easily realized, fourtopics will be used to stress the concept of sequence.* The theoremsfound under each topic, taken in the order given, form a possiblesequence. In these sequences, however, some theorems have proofsnot suitable for examination purposes but are essential as links in achain of theorems. Others have more suitable proofs and these arethe ones that may be called for on the examination. The four topicsare as follows :

a Congruence and parallelismDefinition: Parallel lines are li es which lie in the same

plane and do not intersect, however far they areextended.

Assumed Theorem: Through a given point only onestraight line can be constructed parallel to a given line.

Theorems : (1) When two parallel lines are cut by atransversal the alternate-mterior (orcorresponding) angles are equal.

(2) The sum of the angles of a triangle is equalto a straight angle.

(3) Two triangles are congruent if two anglesand a side opposite one them areequal to the corresponding jearts of theother.

(4) Two right triangles are c ngruent if thehypotenuse and a leg of one are equato the corresponding parts of the other.

A diameter perpendicular to a chord of acircle bisects the chord.

Nork. If in the proof of (4) the hypotenuses are placed together, then insertthe theorems : If two sides of p, triangle are equal the angles opposite thesesides are equal; and if two angles of a triangle are equal the sides oppositethese angles are equal.

* This does not mean that anfy these four topics shall be used to stress sequen-tial thinking- Teachers may =re to consider other - ,.ins of propositions,such as(1) Propositions 2, 14, 15, 16(2) Proposi ions Z 4, 10, 13. 14, 22, 24

MATHEMATICS, GRADES 10, 1 25

b Angle measurementDefinition: A circle is a plane closed curve all points of

which are equally distant from a fixed point.Assumed Theorem: A central angle is measured by its

intercepted arc.Theorems: (1) H two sides of a triangle are equal the

angles opposite these sides are equal.(2) An exterior angle of a triangle is equal

to the sum of the two nonadjacentinterior angles.

An angle inscribed in a circle is measuredby one-half its intercepted arc.

a Case where the center of the circleis on one side of the angle.

b Cases II and III where the center ofthe circle is inside and where it is out-side the angle.

(4a ) An angle formed by two chords inter-secting inside the circle is measured byone-half the sum of the intercepted arcs.

(4b ) An angle formed by two secants inter-secting outside the circle is measured byone-half the difference of the interceptedarcs.

d with either of two different theoi

c SimilarityDefinition: Similar polygons are polygons which have

their corresponding angles equal and their correspond-ing sides proportiona

Assumed Theorem: A line parallel to one side of a tri-angle and intersecting the other two sides divides thesesides proportionally.

Theorems: (I) If two angles of one triangle are equalto two angles of another triangle, thetriangles are similar.

( ) If in a right triangle the altitude is drawnupon the hypotenuse

THE UNIVERSITY 0 THE STATE OF NEW YORK

The two triangles thus formed aresimilar to the given triangle andsimilar to each other.

Each leg of the given triangle is themean proportional between thehypotenuse and the projection ofthat leg on the hypotenuse.

The square of the hypotenuse of a righttriangle is equal to the sum of thesquares of the other two sides.

Norm While (1) is the theorem needed in this sequence, proposition 45(see page 15) of which this should be considered a corollary, will be thereqWred theorem.

d AreaDefinition: The area of any plane surface is the number

of square units of a given kind which it contains.Assumed Theorem: The area of a rectangle is equal to

the product of its base and its altitude.Theorems: (1) The area of a parallelogram is equal to

the product of one side and the altituderawn to that side.

2) The area of a triangle is equal to one-halfthe product of a side and the altitudedrawn to that side.

The area of a regular polygon is equal toone-half the product of its perimeter andits apothem.

The area of a trapezoid is equal to one-halfthe product = of the a titude and the sumof the bases.

(4 ) The area of a ciproduct ofiradius.

NoT This sequence may end with either and (4), or b

7 Although construction methods have been taught in previousyears the teacher should review them. In the tenth year the pupilshould be able to explain why the construction result is valid seenote 5, page

(3b

equal to one,half thecircumference and its

F

M AT TICS, GKADES 10, 11, 12 27

8 Emphasis should be placed on the derivation of the more impor-tant mensuration formulas and on their use. Numerical exercisesfrom various fields and, if possible, of a practical nature should beincluded. The pupil should be able to use competently formulassuch as those given in this list.

9 As in the past one or more original exercises will be calledfor on the final examination. It is, of course, impossible to prescribeall the statements that may be used as authorities to support proofsof original exercises. Such a list depends entirely on the text in use.In general the reasons cited for various steps in a proof should berestricted to the definitions, assumptions, theorems and corollarieswhich are formally set forth in the textbook.

10 Here the emphasis should be placed not so much on the abilityto reproduce the proof of a locus theorem as on the understanding ofits meaning and use. In addition to the two locus theorems (seepropositions 26, 27) other fundamental locus theorems should beconsidered informally and should be used to determine the positionof points by means of intersecting loci. If time permits, the studyshould be extended to include the construction of simple geometricfigures by means of intersecting loci. Simple problems having to do,,with loci expressed algebraically should be included (see illustrationgiven in notes 18-22).

Some teachers may wish to extend the work to three dimensionalloci. If so, the illustrations chosen should be those that follownaturally from the corresponding ideas in plane geometry, forexample : the locus of points in space at a given distance from a givenpoint ; equidistant from two given points.

11 In order that the basic skills of arithmetic be kept in review,it is recommended that mensuration problems be selected whichrequire, occasionally at least, the use of numbers other than simpleintegers_ Furthermore, it should not always be expected that thefinal answer be integral. Care should be taken to see that the datagiven in numerical problems are consistent and that the accuracy orprecision expected in the final answer is stated.

12 A detailed discussion of approximate numbers and of standardpractice with such numbers should not be undertaken at this level.However, if teachers feel that the character of the class warrantsan extension 7.4 this topic beyond that suggested for the ninth grade,they may wish to consider the following points :

a When, for example, it is given that a leis understood that the true length

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amount, but rather that it is greater than 8 feet inchesand less than 8 feet 44 inches. That is to say, the apparenterror (greatest possible error) in this measured length isI of an inch. Similarly, if a weight is given as 6.4 pounds,it is understood that the true weight is between 6.35 poundsand 6.45 pounds and that the apparent error is .05 pounds.

b Measurements given in terms of the same unit are said tohave the same precision if they have the same apparenterror.

c If numbers representing measurements of different precisionare to be added (or subtracted), good practice is toround off whatever numbers are necessary so that theresulting numbers have at most one decimal placemore than the least precise measurement and thenround off the sum (or difference) to the same precisionas the least precise number. For example:

(1) Add 3.492 ft, 7 7 ft, and 6.884 ft3.497.76.88

18.07 18.1 ft(2) Subtract 7.8 in. from 12.437 in,

12.447.84.64 4.6 in.

d The digits 1 through 9 are called significant figures. Alsothe digit 0 may or may not be significant. It is notsignificant when it is used merely to locate the positionof a decimal point. For example :

1 n the number .0024, the 2 and the 4 are significantfigures but the two zeros are not.

(2) In the number 34.08, all the digits including the zeroare significant.

(3) In the number 2400, the 2 and the 4 are significantfigures but we have no way of telling whether thezeros are or are not significant. (If the number werewritten in the form 2.4 X 10°, we would then knowthat the zeros are not significant .)

MATHEMATICS, GRADES 10, 11, 12

e If a number has n significant figures, it is said to haven-place accuracy. Thus, .437 has three-place accuracy ;.00437 also has three-place accuracy ; 0.2057 has four-placeaccuracy ; 24000 has two place accuracy and may havethree, four, or even five-place accuracy.

f If two numbers are to be multiplied (or divided), it is goodpractice to round off the more accurate number and thenround off the product (or quotient) to the same accuracyas the less accurate number. For example :

(1) Mu tiply .02754 by 43.0275

43825

11001.1825 = 1.2

(2) Divide 729.864 by 1355.406 = 5.41

1 5)729.9675

54.954.0

.810g The folio ing rule should be emphasized throughout this

work. All numerical results, before they are stated infinal form, should be obtained with at least one more digitthan the number of significant digits allowed by theapproximate data. Then this last digit is rounded off."(Quoted from the Twelfth Yearbook of the National Coun-cil of Teachers of Mathematics, " Approximate Computa-tion," Aaron Bakst, 1937, page 54.)

13 In accelerated classes it may be desirable to extend this workto include the solution of the quadratic equation by completing thesquare and by means of the quadratic formula.

14 Whenever possible algebraic symbolism should be used as ameans to simplify and clarify the proofs of certain propositions. Thisis especially desirable in connection with angle sum and measure-ment of angles in a circle.

30 THE UNIVERSITY OP THE STATE OF NEW YORK

To illustrateABC and ADE are secants with chord CD drawn. Let the num-

ber of degrees in L CDE be represented by .2- and in L DCA by y.

Then the number of degrees in arc CE is Zz and in are DB is 2y.But the number of degrees in A is (-v y). Therefore an angleformed by two secants intersecting outside the cit _le is measured byone half the difference of the intercepted arcs.

15 In addition to the wide variety of original exercises in geometrywhich can be proved algebraically there are certain geometrictheorems in which algebraic proof can be used to advantage. Seepropositions 49, 50, 56, 57, 58, 59, 62, 63. 64, 66.

16 Although this item is optional, it is suggested, if time permitsthat it be included in the required work of the tenth grade. It isthe only entirely new trigonometric idea proposed for this level and,in addition, exemplifies the integration of geometry, algebra andtrigonometry in the field of area.

17 Coordinate geometry furnishes a new technique whereby it ispossible to establish simple geometric relationships and to provecertain geometric theorems in a much easier manner than that of theusual synthetic method. Special care must be exercised, however,

make sure that proofs by means of coordinate geometry are notcircular.

18 Exercises such as the following should be considerite as an equation the locus -of. points _(1are equal to 4, (2) whose abscissas- a

MATHEMATICS, GRADES 10, 1 1, 12 31

equal, (3) such that the sum (or difference) of thecoordinates is a given constant. Represent these locigraphically.

b Find the area of the triangle whose vertices are A (3, 2),B(8, 3), C(4, 10). Suggestion: Draw- the ordinates ofA,--B and C and use the formula for the area- of a trapezoid.(This method-of-finding areas should be extended to includeother rectilinear figures.)

19 'The formulas .v and y = should be2derived in class and used in exercises such as :

a Given the points A (1, 1), B(10,3), C(12, 10) and D(3,8),show that the line segments AC and BD bisect each other.What kind of quadrilateral is ABCDP

b Find the lengths of the medians of the triangle whose verticesare (2, 8), (10, 12), and (16, 0). (See note 20. )

c Find the coordinates of the point on the x-axis which '-equidistant from the points (2, 3) and (-10, 3)

20 The distance formula d = -Fbe derived in class for the case in which the points are inand used in exercises such as :

a Find the perimeter of the triangle whose vertices are-1, 2), (3, 5), and (0, 1).

b Show that the triangle whose vertices are 1 -2),and (5, 6) is a right triangle.

c Show that the circle whose center is the point (3, 2) andwhich passes through the point (-.$, 2) Will pass through thepoints (3, 4), and (3, 0).

d Show that the locus of paints at a given distance r from theorigin is given by the equation x2 -F y2 r2.

Show that the following points lie on(-- (1, 3), (2, 5)

Y2 shoulduadrant 1,

-a _thight :

The mean ng of slope shoulci be made clear and the formulaY1-77- Y2 -should- be derived- in class and used in a variety of

e ereises. Pupils should know that par inles have the sameslope d, conversely, lines having the same slope are parallel.. Exer-

es such as-the following are -suggested :

a

32 THE UNIVERSITY OP THE STATE OF NEW YORK

a Plot the points A (2, 8), B (6, 4), C(3, -2), and D (-1, 2).Find the slope of line AB, of line DC, of line AD and ofline CB. Why is ABCD a parallelogram ? Prove thatABCD is a parallelogram by showing (1) that the oppositesides are equal, (2) that the diagonals bisect each other.

b Using the formula for the slope of a line, prove that the fol-lowing points lie on the same straight line :

(1) (0, 0), (2, 3), (4, 6)(2) (3, 1), -4, 1), (6, 1)

22 This work should include a study of the families of lines repre-sented by the equations y itiA- and y = x -I- b. Exercises such asthe following should be considered :

a Find the equation of the straight line whose slope is 3 andwhich passes through a point on the y-axis four units abovethe origin.

b Where does the line whose equation is 5 - 2y = 20 e ossthe x-axis ? the y-axis ?

c Express algebraically the locus of point(1) Equidistant from the points (3, 5) and (-1, 5)

Equidisvmt from the points (-1, 6) and (-1, 4)What point satisfies both conditions given in (1) and (2) ?

d Express algebraically the locus of points which are four unitsfrom the line whose equation is y = ; three units fromthe line whose equation is

e The coordinates of the vertices of a quadrilateral are (0, 0),(: 2, 0), (12,5) and (0, 5). Write the equations of thesides of the quadrilateral ; of the diagonals of the quadri-lateral_

The vertmccs of quadrilateral ABCD are -2, -3),B(10,7 C(2, 9) and D(-4, 5). By using the formulafor the slope of a straight line show that the line joiningthe midpoints of AB and BC is parallel to the line joiningthe midpoints of CD and DA.

23 In addition to an understanding and use of the ideas, conceptsand principles of geometry there are other outcomes of a more gen-eral nature which should be recognized as an indispensable part ofthe tenth grade work. Frequently the study of geometry is justifiedon the ground that, perhaps more than any other subjects., it contributesto critical thinking and sound reasoning. This may be so but itdoes not follow automatically. Without a continual and persistent

7

MATHEMATICS, GRADES 10, 11, 12 33

effort on the part of the teacher and the student to correlate thethintEing in geometry with that in nonmathematical fields very little,if any, improvement in the ability to think soundly and critically islikely to result.

Accordingly teachers are urged to return to discussions of thissort again and again in order to make certain that the essential ideasas suggested are reasonably well understood. Ways and means ofpresenting this material will, of course, vary with the individualteacher. Some teachers may prefer to extend this work to includeother ideas not specifically mentioned here. It is not the intent toprescribe exactly what shall be done along these lines. This will bedetermined somewhat by the time element and certainly by the teacher.Also it is not easy to test the progress that pupils make along suchlines. The attempt, however, should be made and questions pertain-ing to such work will be given on the final examination.

Some of the points which may well be stressed are given in theparagraphs that follow :

a The significance of de ration. Pupils should take part inclass discussions which show the necessity of a clearunderstanding of the meaning of terms as a basicrequirement in any intelligent argument. For example,what terms in the following statements require defini-tion if the meaning of the statement is to becomeclear ?

Eligibility for membership on a school team is dep ndenton a satisfactory scholastic standing.

High school fraternities are undemocratic.Argentina is a fascist state.

(2)

b The significance of assuinFtions. Pupils should appreciatethe necessity of assumpt ons as the basis of any argu-ment and should realize that conclusions reached cannot be relied on unless the truth of the assumption isaccepted. A suggestion from the teacher that eachpupil bring to class an example showing how a certaintheory, belief or doctrine is based on one or moreunproved propositions or assurnptions will frequentlyyield most satisfactory results. The following areoffered as suggestions

The Nazi race theory and its results

34 THE UNIVERSITY OP THE STATE OF NEW YORIC

(2) The United States protective tariff and its consequences(3) Isolationism in the United States after World War I and

its effect on the League of Nations

c Converse and inverse Propositions. Pupils should understandthe meaning of such propositions and should realize thatthe converse or the inverse of a proposition is not alwaystrue simply because the direct proposition is true.

To illustrateDirect : In a Bestrite pen the ink flows freely.Converse: If the ink in a pen flows freely, it is a Best-

rite.Inverse If a pen is not a Bestrite, the ink d es not

flow freely.Is the converse true ? The inverse ? Is it safe to reason

from the inverse that it is unwise to buy a pen that is nota Bestrite?

Reasoning from the converse and from the inverse is atrick of the demogogue and the propagandist. An effectiveoutcome of the teaching of geometry should be a realizationof the need to be watchful for fal:-cious reasoning of thiskind when heard on the radio or from the platform or

. .when found in editorials and advertisements .

d Indirect reasoning. I Frequently it is found that indirect proofis difficult for beginners in geometry, ; consequently someteachers may feel inclined not to teach it. It should not,however, be entirely neglected. The mere fact that it isused so commonly in everyday life justifies giving time andthought to the matter. Whether or not indirect proof iscompletely rigorous need not concern us here. In fact,striving for complete rigor throughout the study of geom-etry frequently does more harm than good. If pupils indealing with indirect proof recognize the necessity of con-sidering all the possibilities that can exist in a givensituation and of eliminating all such possibilities exceptthe one which they wish to establish as true, the objectiveof teaching indirect proof has, in large part, been accom-plished.

Propositions 8 and 9, either of which may be proved bythe indirect method, and many original exercises such, for

ample, as the diagonals of a trapezoid can not bisect

MATHEMATICS, GRADES 10, 11,

each other, illustrate the fact that at times the indiremethod of proof is to be preferred. Much of its value,however, will be lost if the idea is not carried over intolife situations.

e circular reasoning_ By giving due attention to sequence ingeometry it is likely that the meaning of circular reasoningcan be made clear. Instances from geometry in whichreasoning is circular should be cited and the pupil shouldbe made to see clearly why conclusions resulting fromcircular reasoning are invalid. Point out, for example, thecircular reasoning involved in proving both propositions8 and 9 by the indirect method in which each is made todepend on the other. Encourage pupils to bring to classinstances in daily life in which circular reasoning is used.

f Use of the terms determined," underdetermined" andoverdetermined." These mathematical ideas have their

counterpart in every day affairs and should -be emphasized.One of the most common mistakes made by beginners ingeometry is the use of overdeterrnined lines (See note 5).Class discussions such as those listed below tend to clarifythese ideas

(1) In which of the following cases are the values of x and ydetermined, underdetermined, or overdetermined?

Y

x :y =2x 2y =

(b) x± y=3 x ± 2y = 21

(2 ) A telephone number is Exeter 5094 R. Which part ofthe nuMber denotes the exchange? The line ? Theparty un the line? Does this number represent asituation which is determined, overdetermined, orunderdeterrnined?

reievancy. Frequently an argument heard on the platformor in the courtroom, or in ar.y. place where it is thepurpose ,to persuade, irrelevant statements are deliber-ately injected in order to confuse the issue. Extrava-

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gant advertising also often makes use of this device.If pupils are encouraged to look for this same featurein their problems in geometry it may rightly beexpected that they may become more critical of whatthey see and hear. Problems such as the followingmay help.

In which of the foll wing statements is there moreinformation given in the hypothesIs than is necessaryto reach the conclusion?

Line segments joining the midpoints of the oppositesides of a parallelogram b:sect each other_

If a parallelogram is circumscribed about a circle, theparallelogram is a rhombus.

The altitude upon the base of an isosceles right trianglebisects the vertex angle.

(4) If a piece of metal is pure iron, a magnet will attract it.

MATHEMATICS, GRADES 10, 11, 12

SUGGESTED TIME SCHEDULE AND TEACHINGEQUENCE

-37

TimeAllotmentUnit Topics in Days

I Understanding and use of basic terms and con-cepts ; significance of definition in mathe-matical and nonmathematical settings. 12-14

II Understanding and use of axioms and postulates;meaning and importance of assumptions innonmathematical situations.

III Congruence. See propositions 1-3. 6-7IV Fiindamental constructions ; meaning of the

terms determined, underdetermined, and over-determined in both geometric and nongeo-metric fields.

V Parallel and perpendicular lines. See proposi-tions 11-14 ; meaning and use of indirectreasoning in both geometry and everyday life.

VI Angle sum. See proposition 22-25. Converseand inverse propositions.

VII Parallelograms. See proposition 15-21. Squential thinking ; circular reasoning.

VIII Loci and construction ; coordinate geometpage 19, VI, AC. 15-17

IX Circles. See proposition 30,38. 8-9X Measurement of angles in a circle. See proposi-

tions 39-42.Similarity. See propositions 43-54-Numerical trigonometry. See page 19, V, ADArea. See propositions 55-59. Coordinate

geometry. See page 19, VI, DP. Numericaltrigonometry. See page 19, V, P.

Regular polygons and the measurement of thecircle. See propositions 6 umerical

gonometry. See page 19, V,

8-10